tensors over 
’
Abstract
In this note, we correctly determine the orbits of the (maximum) rank nine tensors under the action of , the semi-direct product of (a direct product of) general linear groups with the symmetric group on five elements. Under this group action, there are two orbits and hence, two canonical forms.
AMS Subject Classifications:
1. The orbits of the rank nine tensors under the 
group action
In [Citation1], it is shown that the group action of partitions the set of (maximum) rank nine tensors into two disjoint orbits
and
, where
and
. This is correct. We then calculated the group action of
on
and
separately, which is not the correct way to analyze the
group action on the set of rank nine tensors. This gave us
sets (Tables 2–9 in [Citation1]), which we incorrectly asserted as being the orbits of the
group action.
In this note, we give the correct number of orbits of the rank nine tensors (under ) and their corresponding canonical forms. In [Citation1], the canonical forms for
and
under
were correctly determined to be
The action of should reveal that
and
either remain disjoint or else they combine to a single orbit (under
). Tables 2–9 in [Citation1] are not needed since they correspond to the 261 irrelevant sets described above. We now apply the group action of
on
, obtaining
(not necessarily distinct). This gives us the orbit of
under
, namely
We check that none of the elements of belong to
. Thus,
and
are the two disjoint orbits of the rank nine tensors of format
over
under the action of
. The canonical forms for
and
under
are
Acknowledgements
The authors are grateful for the valuable comments made by the referee.
Notes
No potential conflict of interest was reported by the authors.
References
- Stavrou SG, Low RM, Hernandez NJ. Rank classification of 2 × 2 × 2 × 2 × 2 tensors over F2. Linear Multilinear Algebra. 2016;64(11):2297–2312.